We develop an inductive approach to the representation theory of the Yokonuma–Hecke algebra Yd,n(q), based on the study of the spectrum of its Jucys–Murphy elements which are defined here. We give ...explicit formulas for the irreducible representations of Yd,n(q) in terms of standard d-tableaux; we then use them to obtain a semisimplicity criterion. Finally, we prove the existence of a canonical symmetrising form on Yd,n(q) and calculate the Schur elements with respect to that form.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We consider the construction of refined Chern–Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the ...stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on C2. Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the mn shuffle conjecture.
On considere la construction des invariants de Chern–Simons rafinés des noeuds toriques par M. Aganagic et S. Shakirov du point de vue DAHA de I. Cherednik. On démontre une conjecture de Cherednik sur la stabilisation des superpolynômes, et puis on utilise les resultats de O. Schiffmann et E. Vasserot pour relier les invariants des noeuds au schéma d'Hilbert des points sur C2. Ensuite on utilise les techniques du deuxième auteur pour calculer ces invariants explicitement dans le cas non-coloré. On propose aussi une conjecture reliant ces constructions à l'algèbre de Cherednik rationelle, comme dans l'article du premier auteur avec A. Oblomkov, J. Rasmussen et V. Shende. Parmi les conséquences combinatoires de cette étiude on a énoncé de la conjecture de mn battage.
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We prove that cyclotomic Yokonuma–Hecke algebras of type A are cyclotomic quiver Hecke algebras and we give an explicit isomorphism with its inverse, using a similar result of Brundan and Kleshchev ...on cyclotomic Hecke algebras. The quiver we use is given by disjoint copies of cyclic quivers. We relate this work to an isomorphism of Lusztig.
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In recent years, there has been an increased interest in exploring the connections between various disciplines of mathematics and theoretical physics such as representation theory, algebraic ...geometry, quantum field theory, and string theory. One of the challenges of modern mathematical physics is to understand rigorously the idea of quantization. The program of quantization by branes, which comes from string theory, is explored in the book. This open access book provides a detailed description of the geometric approach to the representation theory of the double affine Hecke algebra (DAHA) of rank one. Spherical DAHA is known to arise from the deformation quantization of the moduli space of SL(2,C) flat connections on the punctured torus. The authors demonstrate the study of the topological A-model on this moduli space and establish a correspondence between Lagrangian branes of the A-model and DAHA modules. The finite-dimensional DAHA representations are shown to be in one-to-one correspondence with the compact Lagrangian branes. Along the way, the authors discover new finite-dimensional indecomposable representations. They proceed to embed the A-model story in an M-theory brane construction, closely related to the one used in the 3d/3d correspondence; as a result, modular tensor categories behind particular finite-dimensional representations with PSL(2,Z) action are identified. The relationship of Coulomb branch geometry and algebras of line operators in 4d N = 2* theories to the double affine Hecke algebra is studied further by using a further connection to the fivebrane system for the class S construction. The book is targeted at experts in mathematical physics, representation theory, algebraic geometry, and string theory. This is an open access book.
In this paper we establish affinizations and R-matrices in the language of pro-objects, and as an application, we construct reflection functors over the localizations of quiver Hecke algebras of ...arbitrary finite types. This reflection functor categorifies the braid group action on the half of a quantum group and the Saito reflection.
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For any affine Weyl group, we introduce the pre-canonical bases. They are a set of bases {Ni}1≤i≤m+1 (where m is the height of the highest root) of the spherical Hecke algebra that interpolates ...between the standard basis N1 and the canonical basis Nm+1. The expansion of Ni+1 in terms of the Ni is in many cases very simple and we conjecture that in type A it is positive.
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We investigate deformations of skew group algebras arising from the action of the symmetric group on polynomial rings over fields of arbitrary characteristic. Over the real or complex numbers, ...Lusztig's graded affine Hecke algebra and analogs are all isomorphic to Drinfeld Hecke algebras, which include the symplectic reflection algebras and rational Cherednik algebras. Over fields of prime characteristic, new deformations arise that capture both a disruption of the group action and also a disruption of the commutativity relations defining the polynomial ring. We classify deformations for the symmetric group acting via its natural (reducible) reflection representation.
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We introduce a family of quiver Hecke algebras which give a categorification of quantum Borcherds algebra associated to an arbitrary Borcherds-Cartan datum.
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We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras ...under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at 1 can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we recover the results of Aubert-Michel-Rouquier and Pan on the explicit description of theta correspondence between Harish-Chandra series.
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